Melnikov transforms, Bernoulli bundles, and almost periodic perturbations. (English) Zbl 0707.34041

Meyer, Kenneth R.; Sell, George R.

Trans. Am. Math. Soc. 314, No. 1, 63-105 (1989).

The authors make a very competent study of nonlinear time-varying perturbations of an autonomous vector field in the plane \(R^ 2\). The main hypothesis is that the unperturbed equation has a homoclinic orbit. Using a generalization of the Melnikov method and the concept of a skew product flow, the authors show that the perturbed equation has a transversal homoclinic trajectory. Also, a study of the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow is maked.
The properties of the skew product flow are discussed in a greater detail in § 2. The main result of § 3 is that if the Melnikov transform has a simple zero set, then there exists a normally hyperbolic homoclinic bundle which, together with a suitable shadowing lemma, will beget the Bernoulli bundle (a fiber bundle with fiber maps which are Bernoulli automorphisms). § 4 contains the generalization of hyperbolic invariant set for skew product flows and a proof of a generalization of the shadowing lemma. The shadowing lemma is the main tool used to prove the existence of the Bernoulli bundle invariant set. An illustration of this study is given in § 6, exactly, the perturbed Duffing equation with negative linear stiffness.

Reviewer: C.Simirad

MSC:

34D30	Structural stability and analogous concepts of solutions to ordinary differential equations
54H20	Topological dynamics (MSC2010)
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C55	Periodic and quasi-periodic flows and diffeomorphisms

Keywords:

nonlinear time-varying perturbations; autonomous vector field; homoclinic orbit; Melnikov method; skew product flow; shadowing lemma; Bernoulli bundle invariant set; perturbed Duffing equation